This paper develops a general approach for deep learning for a setting that includes nonparametric regression and classification. We perform a framework from data that fulfills a generalized Bernstein-type inequality, including independent, $φ$-mixing, strongly mixing and $\mathcal{C}$-mixing observations. Two estimators are proposed: a non-penalized deep neural network estimator (NPDNN) and a sparse-penalized deep neural network estimator (SPDNN). For each of these estimators, bounds of the expected excess risk on the class of Hölder smooth functions and composition Hölder functions are established. Applications to independent data, as well as to $φ$-mixing, strongly mixing, $\mathcal{C}$-mixing processes are considered. For each of these examples, the upper bounds of the expected excess risk of the proposed NPDNN and SPDNN predictors are derived. It is shown that both the NPDNN and SPDNN estimators are minimax optimal (up to a logarithmic factor) in many classical settings.

翻译：本文针对包含非参数回归与分类的场景，提出了一种深度学习的通用方法。我们构建了一个适用于满足广义伯恩斯坦型不等式的数据框架，涵盖独立观测、$φ$混合、强混合及$\mathcal{C}$混合观测。提出了两种估计器：无惩罚深度神经网络估计器（NPDNN）与稀疏惩罚深度神经网络估计器（SPDNN）。针对这两种估计器，我们建立了在Hölder光滑函数类与复合Hölder函数类上的期望超额风险界。研究涵盖独立数据以及$φ$混合、强混合、$\mathcal{C}$混合过程的应用场景。针对每种案例，推导了所提出的NPDNN与SPDNN预测器期望超额风险的上界。研究表明，在众多经典设定中，NPDNN与SPDNN估计器均达到极小化极大最优性（至多相差对数因子）。